Static analysis of hydraulic systems  Library of hydraulic elements and math models
Library of hydraulic elements and their mathematical models
Static calculation is a basic stage at designing hydraulic drives and transmissions of various machines and mechanisms. On a basis of results of static calculation synthesis of hydraulic circuit and selection of elements of the certain standard size is carried out. But at the same time it is closely connected with dynamic analysis as preliminary stage necessary for the correct giving of an initial condition of hydraulic system at simulating of transient processes.
In resulted below equations of statics of hydraulic elements the same designations are accepted, and for the description of the input data of elements the same identifiers and tables are used, as at the dynamic analysis.
Pump. For the pump description it is enough to write down equation of a shaft moments (node k) and equations of flows in input (node i) and output (node j) in view of volumetric losses. Thus nonuniformity of the pump flow owing to kinematic features and compressibility of liquid in sucking and pressure cavities is not considered. In view of the accepted assumptions the mathematical model of the pump looks like [1, 2]:
, (2)
where q_{p }– maximal geometric volume of pump; f (q) – parameter of pump geometric volume regulation; – 1≤ f (q) ≤ 1; ω_{s} – angular speed of engine (diesel engine) shaft; а_{ω} , а_{р} , а – coefficients of pump hydro mechanical losses depending on angular speed, pressure, and constant value of hydro mechanical losses; u_{e }– transfer number of gear between engine and pump; k_{lea}– coefficient of pump volumetric losses; for Q_{i} , p_{i} the sign "plus" is accepted, for Q_{j} , p_{j}– "minus".
Values of а_{ω} , а_{р} , а , k_{lea } are chosen from the catalogue or from passport characteristics of mechanical and volumetric efficiency of the certain standard size pump.
The hydro mechanical losses depending on pressure are calculated modulo for opportunity of consideration of brake modes and reversing flow (when f (q)<0).
Hydraulic motor. The mathematical model of hydraulic motor contains equation of moments in node k, and also equations of flows in input (node i) and output (node j) in view of volumetric losses. Without taking into account nonuniformity of flow (it is similar to pump) the system of equations of hydraulic motor looks like [1, 2]:
(3)
where ω_{k} – angular speed of hydraulic motor shaft; q_{м}_{ }– maximal geometric volume of hydraulic motor ; f (q) – parameter of hydraulic motor geometric volume regulation; – 1≤ f (q) ≤ 1; М_{l} – loading moment; b_{ω} , b_{р} , b – coefficients of hydraulic motor hydro mechanical losses depending on angular speed, pressure, and constant value of hydro mechanical losses; u_{mech} – transfer number of the working mechanism gear; k_{lea}– coefficient of volumetric losses of hydraulic motor; for Q_{i} , p_{i} the sign "plus" is accepted, for Q_{j} , p_{j}– "minus". As well as for pump, values b_{ω} , b_{р} , b , k_{lea} are chosen from the catalogue or from passport characteristics of mechanical and volumetric efficiency of the certain standard size hydraulic motor. Hydro mechanical losses in the equation of moments are written down in view of direction of shaft rotation (sign ω_{k}) and opportunities of consideration of a brake mode  p_{i} – p_{j} .
Hydraulic cylinder. Hydraulic cylinder is described by equation of balance of forces (forces of pressure, external loading, forces of friction) at piston progressive motion (node k) and equations of flows in input (node i) and output (node j). On a basis of standard assumption about absence of leakages in hydraulic cylinder with rubber and other soft seals equations of statics of hydraulic cylinder look like [1, 2]:
(4)
where v_{k}_{ } – speed of piston motion; F_{i} = π (D– D)/4 – piston working area in cavity I adjoining node i (here D_{c} – cylinder diameter of; D_{i} – rod diameter in cavity I ); F_{j} = π (D– D)/4 – piston working area in cavity II adjoining node j (here D_{j}_{ } – rod diameter in cavity II ); h – coefficient of viscous friction; R – force of friction in seals at absence of pressure; R_{c}– force to rod. Coefficients of proportionality between pressures in cavities I (node i) and II (node j) and force of friction in seals
k= π f (D_{c} + D_{i }) H / 2, k= π f (D_{c} + D_{j} ) H / 2. (5)
Here f – coefficient of friction of seal material and cylinder wall; Н – height of seal.
Pipeline. Mathematical model of pipeline with liquid consists of equality of flows in input (node i) and output (node j) and equations of losses of pressure on length and looks like [1, 2]:
(6)
where – coefficient of pressure losses on length of pipeline,
(7)
here Re = 4  Q_{i}  / (d) – Reynolds’ number, – kinematic viscosity of liquid; – density of working liquid; d_{p }and L_{p }– diameter and length of pipeline.
Deadlock site of pipeline (cavity). For deadlock site of pipeline losses of pressure on length can be neglected, and then equations of statics of deadlock site of pipeline become:
(8)
Local resistance (throttle). Flow of liquid through throttle is connected with pressure difference in input (node i) and output (node j) by the known dependence [1, 2]:
(9)
where – flow coefficient, = (here – coefficient of hydraulic resistance); – area of through passage section of throttle.
Tee (a divider or the summer of flows). Equations of tee flows in nodes i, j, k at division of flow look like [1, 2]:
(10)
where – flow coefficients in tee branches i – j , i – k ; = (here – coefficients of hydraulic resistances of tee branches i – j , i – k ); – areas of through passage sections of tee in nodes j and k .
Equations of tee flows at summation of flows are similar (10), but have other values of flow coefficients. The assumption is accepted, that coefficients of hydraulic resistances at change of flow direction do not vary.
Valves.
Valve of direct action is described by equations of statics of lockingregulating element (node k) and flows in nodes i and j [1, 2]:
(11)
where F_{i } and F_{j} – working areas of the valve lockingregulating element from pressure head and drain line; с – rigidity of spring; – value of spring preliminary compression; – stroke of lockingregulating element; – area of through passage section of throttle connected in parallel to the valve; – average diameter of throttling crack of the valve; – angle of the valve cone.
The resulted equations concern to pressure relief and check valves. The corresponding equations for reducing valve have insignificant differences. In equations (11) hydro dynamical force is not considered.
Pilot controlled valve (of indirect action) consists of two elements: the main valve with nodes r, s, t and the pilot valve with nodes i, j, k. If a discharge node j is general for both valves, then s = j. Mathematical model of statics of pilot controlled valve looks like [2]:
(12)
where – working areas of auxiliary valve lockingregulating element from pressure head and drain line; – working areas of basic valve lockingregulating element from pressure head line and cavity between valves; с and С – rigidities of springs of auxiliary and basic valves; preliminary compressions of springs of auxiliary and basic valves;stroke of mobile parts of auxiliary and basic valves; G – conductivity of orifice apertures of basic valve; average diameters of throttling cracks of auxiliary and basic valves; angles of cone of auxiliary and basic valves;
(13)
Hydraulic accumulator. For the description of stationary mode of the hydro pneumatic or spring accumulator it is necessary to write down equations of balance of its piston (membrane) in node k, equation of flow in input (node i) and equation of polytropic process in gas cavity (node j) [1, 2]:
(14)
where F = D/ 4 – working area of piston; D – diameter of piston; с – rigidity of spring;– preliminary compression of spring; V – general volume of hydraulic accumulator; – pressure of charged gas; n – exponent of polytropic process; – atmospheric pressure.
Power regulator is intended for maintenance in the pump certain working range of constancy of the power taken away engine: pQ = const. Actually, power regulator provides a constancy: p f(q) = const. But, Q = q_{p} f(q)where q_{p}is practically constant, so it is possible to consider that power regulator provides constant taking of power away engine. The static characteristic of power regulator (Fig. 2) looks like piecewise linear function approximating hyperbolic dependence of the pump geometric volume versus pressure: p f (q) = const. In practice it is carried out by means of choosing of springs for the 1st and the 2nd branches of power regulator characteristic (accordingly, AO and ОD, Fig. 2).
Fig. 2. Static characteristic of power regulator
Then in statics power regulator is described by the following equation [1, 2]:
(15)
where А – coefficient considering for axialpiston pumps the additional moment, acting to swinging unit; F – working area of plunger under pressure of each of two pipelines;– force of preliminary compression of spring; R_{fr} – force of friction; – rigidities of springs; – stroke of plunger on the 1st branch of power regulator characteristic; – maximal stroke of power regulator plunger.
Diesel engine with a centrifugal regulator. Diesel engine with a centrifugal regulator in statics can be described by equation of engine shaft moments (node j) and equation of regulator muff balance (node k) [1, 2]:
(16)
where – diesel engine characteristic at minimal fuel consumption, approximated by finite set of points – increment of torque moment at maximal fuel consumption; – constant parameters of diesel engine regulator adjustment; – loading moment of pump reduced to diesel engine shaft; – transfer ratio of regulator drive; с, F – rigidity and force of preliminary compression of spring; – maximal stroke of regulator muff.
Wheel (wheel carrier). For carrying out of traction calculations of hydro volumetric transmissions of selfpropelled wheel machines it is necessary to consider as one of base elements a wheel (wheel carrier) – Fig. 1. Indexes i, j, k designate accordingly nodes in input (power shaft of wheel), output (point of contact of wheel with road) and machine movement. The model of wheel carrier considered here describes rigid communication of wheel with the hydraulic motor. In view of the accepted assumptions the static mathematical model of wheel (wheel carrier) looks like:
(17)
where М _{i} – wheel moment in view of losses in gear; М _{n} – moment reduced to shaft of hydraulic motor; – wheel traction reaction (circular force); r – dynamic radius of wheel; – efficiency and transfer number of wheel gear; angular speeds of hydraulic motor shaft and wheel; W – total force of resistance to machine moving; N – number of driven wheels (axes).
In established mode circular force R of wheel is connected with relative slipping by dependence [1, 3]:
(18)
where
(19)
Fig. 3. Function of wheel slipping.
Here f ( δ ) – function of wheel slipping (Fig. 3); ω – angular speed of wheel; v – speed of machine progressive movement (node k, Fig. 1). Value of wheel radius r depends on static deflection of wheel under loading:
(20)
where – free radius of wheel; component of machine weight, falling axis; radial rigidity of tire.
